1. Field of the Invention
The invention relates to a method of determining spectrum parameters of a spectrum relates to spectroscopic signals, in which method the spectroscopic signals derived from a substance are sampled in order to obtain sampling values and are approximated by a complex model function which contains the spectrum parameters and exponentially damped sinusoids, in which method there is also made an initial estimate, based on the sampling values, for at least one of the spectrum parameters, the at least one and further spectrum parameters being accurately estimated from the model function by iteration with a least-squares optimisation procedure, and prior knowledge is introduced into the model function.
The invention also relates to a device for determining spectrum parameters of a spectrum related to spectroscopic signals, which device comprises means for generating the spectroscopic signals in a substance, sampling means for obtaining sampling values from the spectroscopic signals, and displaying means for displaying the spectrum, and also comprises programmed means for making an initial estimate for at least one spectrum parameter, which programmed means also comprise a model function of exponentially damped sinusoids and are suitable for storing prior knowledge of the spectroscopic signals and for executing a least-square optimisation procedure on the basis of the sampling values, the programmed means furthermore comprising iteration means for accurately estimating, the at least one and further spectrum parameters by means of the least-squares optimisation procedure.
2. Description of the Prior Art
A method of this kind is inter alia suitable for signals which are mainly exponentially damped, for example for signals obtained during a magnetic resonance experiment from an entire body as the substance as well as from a part of the body. The method can also be used, for example for X-ray spectroscopy or FT infrared spectroscopy.
A method of this kind is disclosed in an article by J. W. C. van der Veen et al, "Accurate Quantification of in Vivo .sup.31 P NMR Signals Using the Variable Projection Method and Prior Knowledge", Magnetic Resonance in Medicine, Vol. 6, No. 1, January 1988, pp. 92-98. The spectrum parameters of a spectrum related to spectroscopic signals (signals obtained during an NMR experiment) are derived via the sampling values (represented in a time domain, which is in contrast with the spectrum which is represented in a frequency domain), directly in the time domain. A least-squares optimisation procedure is used for fitting the sampling values to a model function of spectroscopic signals, in which procedure prior knowledge concerning spectral components can be introduced. The method can in principle be used for an arbitrary model function. In said article, for example a model function is assumed which contains exponentially damped sinusoids. By operation in the time domain, arbitrary parts of the spectroscopic signals can be omitted without giving rise to serious fitting problems. For example, when in the case of NMR the substance contains immobile nuclei and mobile nuclei, the sampling values (of a quickly exponentially decaying spectroscopic signal) originating from the immobile nuclei can be simply omitted. The sampling values of the tail of an exponentially decaying spectroscopic signal can also be omitted. Thus, convolution effects associated with a transformation to the frequency domain in conjunction with weighting, such as base line (wide background in the spectrum due to quickly decaying signals from the immobile nuclei) and line-shaped distortions (due to phase shifts) can be avoided. Fitting is performed partly non-iteratively in order to obtain an initial estimate for at least one of the spectrum parameters used as initial values for further iterative fitting. The non-iterative fitting is comparatively fast and the further iterative fitting enables the method to be used successfully also in the case of a poor signal-to-noise ratio. For parameters which occur linearly in the model function, such as complex amplitudes, no initial values are required, resulting in a faster method. Said article describes the method inter alia for a model function of exponentially damped sinusoids in which parameters such as amplitude, damping factor and frequency occur. Using the least-squares optimisation procedure, the sampling values are iteratively fitted as well as possible to the model function, initial values being required for the damping factor and frequencies which occur as non-linear parameters in the model function. The prior knowledge may comprise a time shift of an instant of a first sample with respect to a time origin and predetermined phases. The least-squares optimisation procedure can be written in the form of a matrix. This is represented by the formule 4 on page 94 of said article, showing a least-squares solution for the amplitude. All terms in the inverse matrix occurring therein are successively calculated, like the terms in the product of the Hermitic matrix with the data vector. In the matrix form a large number of product terms occur, so that a substantial amount of computation time is required for executing the method by means of a computer; this aspect becomes more significant in the case of a comparatively large number of sampling values.